[This post replaces the previous version from October 12, 2008. The new argument is significantly simpler than the one originally posted.]
I want to present here a Calculus II-level proof that the method of partial fractions decomposition works. I will actually show a more general result, which will greatly simplify the presentation and get rid of most problems of the somewhat awkward formulation below.
We need some notation:
- means .
- , , etc, denote polynomials with real coefficients.
The proof that I show below is algorithmic in nature, meaning that it provides us with a method to find the relevant constants for any given specific polynomials and . The constants we obtain are real numbers.
That the method of partial fractions decomposition works means that we can always find the relevant constants the method requires.